Bertil Magnusson - NORDTEST Guide on uncertainty - 2003

advertisement
TR 537
Approved 2003-05
HANDBOOK
FOR
CALCULATION OF
MEASUREMENT UNCERTAINTY
IN
ENVIRONMENTAL LABORATORIES
Bertil Magnusson
Teemu Näykki
Håvard Hovind
Mikael Krysell
Published by Nordtest
Tekniikantie 12
FIN–02150 Espoo
Finland
Phone: + 358 9 455 4600
E-mail: [email protected]
Fax: + 358 9 455 4272
Internet: www.nordtest.org
NT TECHN REPORT 537
Approved 2003-05
Authors:
Bertil Magnusson1
Teemu Näykki2
Håvard Hovind3
Mikael Krysell4
NORDTEST project number: 1589-02
Institution:
SP, Sweden, 2) SYKE, Finland, 3) NIVA, Norway, 4) Eurofins A/S,
Denmark
1)
Title (English):
Title (Original):
Handbook for Calculation of Measurement Uncertainty in Environmental
Laboratories
Abstract:
This handbook is written for environmental testing laboratories in the Nordic countries, in order to
give support to the implementation of the concept of measurement uncertainty for their routine
measurements. The aim is to provide a practical, understandable and common way of
measurement uncertainty calculations, mainly based on already existing quality control and
validation data, according to the European accreditation guideline /12/, the Eurolab Technical
Report No. 1 /3/ and the ISO/DTS 21748 Guide /8/. Nordtest has supported this project
economically in order to promote and enhance harmonisation between laboratories on the Nordic
market.
Practical examples, taken directly from the everyday world of environmental laboratories, are
presented and explained. However, the approach is very general and should be applicable to most
testing laboratories in the chemical field.
The handbook covers all steps in the analytical chain from the arrival of the sample in the
laboratory until the data has been reported. It is important to notice that vital parts of the total
measurement uncertainty are not included, e.g. sampling, sample transportation and possible
gross errors during data storage/retrieval.
The recommendations in this document are primarily for guidance. It is recognised that while the
recommendations presented do form a valid approach to the evaluation of measurement
uncertainty for many purposes, other suitable approaches may also be adopted – see references
in Section 9. Especially the EURACHEM/CITAC-Guide /2/ is useful in cases where sufficient
previous data is not available, and therefore the mathematical analytical approach according to
GUM /1/ with all different steps is to be used.
Basic knowledge in the use of quality control and statistics is required. In order to make it possible
for the reader to follow the calculations, some raw data is given in appendices.
Technical Group: Expert Group Quality and Metrology
ISSN: 0283-7234
Language: English
Class (UDC): 620.1
Key words:
Distributed by:
NORDTEST
Tekniikantie 12
FIN-02150 ESPOO
Finland
Publication code:
Pages: 41
handbook, laboratories, environmental, testing,
measurements, uncertainty, chemical
Report Internet address:
http://www.nordtest.org/register/techn/tlibrary/tec537.pdf
Handbook
for
Calculation of
Measurement Uncertainty
in
Environmental Laboratories
Version 1.2 June 2003
Nordtest project 1589-02
Project participants
Bertil Magnusson, SP, Sweden
Teemu Näykki, SYKE, Finland
Håvard Hovind, NIVA, Norway
Mikael Krysell, Eurofins A/S, Denmark
Drawings by Petter Wang, NIVA, Norway
Valuable comments on the
contents have been provided by:
Rolf Flykt, Sweden
Irma Mäkinen, Finland
Ulla O. Lund, Denmark
Steve Ellison, UK
Contents:
1
2
DEFINITIONS AND ABBREVIATIONS ...................................................... 1
INTRODUCTION............................................................................................ 3
2.1
2.2
2.3
3
CALCULATION OF EXPANDED UNCERTAINTY, U - OVERVIEW ...... 6
3.1
3.2
3.3
4
CERTIFIED REFERENCE MATERIAL......................................................................... 15
INTERLABORATORY COMPARISONS ........................................................................ 17
RECOVERY ............................................................................................................. 18
REPRODUCIBILITY BETWEEN LABORATORIES, SR ........................... 19
6.1
6.2
7
CUSTOMER DEMANDS ............................................................................................ 10
CONTROL SAMPLE COVERING THE WHOLE ANALYTICAL PROCESS.......................... 10
CONTROL SAMPLE FOR DIFFERENT MATRICES AND CONCENTRATION LEVELS ........ 11
UNSTABLE CONTROL SAMPLES............................................................................... 12
METHOD AND LABORATORY BIAS – u(BIAS) ................................... 15
5.1
5.2
5.3
6
CUSTOMER NEEDS .................................................................................................... 7
FLOW SCHEME FOR UNCERTAINTY CALCULATIONS .................................................. 7
SUMMARY TABLE FOR UNCERTAINTY CALCULATIONS ............................................. 9
REPRODUCIBILITY WITHIN-LABORATORY - u(R W) .......................... 10
4.1
4.2
4.3
4.4
5
SCOPE AND FIELD OF APPLICATION .......................................................................... 3
COMMENT TO CUSTOMERS ....................................................................................... 3
ABOUT MEASUREMENT UNCERTAINTY.................................................................... 4
DATA GIVEN IN STANDARD METHOD ...................................................................... 19
DATA FROM INTERLABORATORY COMPARISONS .................................................... 19
EXAMPLES................................................................................................... 21
7.1
7.2
7.3
7.4
AMMONIUM IN WATER ........................................................................................... 21
BOD IN WASTEWATER ........................................................................................... 21
PCB IN SEDIMENT .................................................................................................. 25
CONCENTRATION RANGES ...................................................................................... 28
8
REPORTING UNCERTAINTY.................................................................... 30
9
REFERENCES............................................................................................... 32
10
APPENDICES................................................................................................ 33
Appendix 1: Empty flow scheme for calculations.................................................. 33
Appendix 2: Empty summary table ........................................................................ 34
Appendix 3: Error model used in this handbook .................................................... 35
Appendix 4: Uncertainty of bias for NH4-N in section 3.2..................................... 36
Appendix 5: Raw data for NH4-N in section 4.3 .................................................... 37
Appendix 6: Raw data for oxygen in Section 4.4 ................................................... 39
Appendix 7: Raw data for BOD in example 6.1..................................................... 40
Appendix 8: Estimation of standard deviation from range ..................................... 41
1 Definitions and abbreviations
s
An estimate of the population standard deviation σ from a limited
number (n) of observations (xi)
x
Mean value
u(x)
Individual standard uncertainty component (GUM, /1/).
uc
Combined standard uncertainty (GUM, /1/)
U
Expanded combined uncertainty close to 95 % confidence interval
r
Repeatability limit – performance measure for a test method or a
defined procedure when the test results are obtained under
repeatability conditions.
Repeatability conditions: Conditions where independent test
results are obtained with the same method on identical test items
in the same laboratory by the same operator using the same
equipment within short intervals of time.
Repeatability (precision under repeatability conditions) is also
sometimes called “within run precision” (ISO 3534-1, /6/).
sr
Repeatability standard deviation of a measurement (can be
estimated from a series of duplicate analyses)
R
Reproducibility limit – performance measure for a test method or
procedure when the test results are obtained under
reproducibility conditions.
Reproducibility conditions: Conditions where test results are
obtained with the same method on identical test items in different
laboratories with different operators using different equipment.
Reproducibility (precision under reproducibility conditions) is
also sometimes called “between lab precision” (ISO 3534-1, /6/).
sR
Reproducibility standard deviation of a measurement (can be
estimated from validation studies with many participating
laboratories or from other interlaboratory comparisons e.g.
proficiency testing data)
Note: R = 2.8 ⋅ s R
Rw
Within-laboratory reproducibility = intermediate measure between
r and R, where operator and/or equipment and/or time and/or
calibration can be varied, but in the same laboratory. An
alternative name is intermediate precision
sRw
Reproducibility within-laboratory standard deviation (can be
estimated from standard deviation of a control sample over a
certain period of time, preferably one year)
Page 1 of 41
CRM
Certified Reference Material
Certified
value
Assigned value given to a CRM, quantified through a certification
process (traceable to SI-unit and with a known uncertainty)
Nominal
value
Nominal value is the assigned value, e.g. in an interlaboratory
comparison where it is the organiser’s best representation of the
“true value”
u(Cref)
Uncertainty component from the certified or nominal value
bias
Difference between mean measured value from a large series of
test results and an accepted reference value (a certified or nominal
value). The measure of trueness is normally expressed in term of
bias.
Bias for a measurement, e.g. for a laboratory or for an analytical
method..
Uncertainty component for bias. The u(bias), is always included
in the measurement uncertainty calculations
u(bias)
RMSbias
∑ (bias
i
)2
n
Interlaboratory General term for a collaborative study for either method
comparison
performance, laboratory performance (proficiency testing) or
material certification.
Page 2 of 41
2 Introduction
2.1 Scope and field of application
This handbook is written for environmental testing laboratories in the Nordic
countries, in order to give support to the implementation of the concept of
measurement uncertainty for their routine measurements. The aim is to provide a
practical, understandable and common way of measurement uncertainty
calculations, mainly based on already existing quality control and validation data,
according to the European accreditation guideline /12/, the Eurolab Technical
Report No. 1 /3/ and the ISO/DTS 21748 Guide /8/. Nordtest has supported this
project economically in order to promote and enhance harmonisation between
laboratories on the Nordic market.
Practical examples, taken directly from the everyday world of environmental
laboratories, are presented and explained. However, the approach is very general
and should be applicable to most testing laboratories in the chemical field.
The handbook covers all steps in the analytical chain from the arrival of the sample
in the laboratory until the data has been reported. It is important to notice that vital
parts of the total measurement uncertainty are not included, e.g. sampling, sample
transportation and possible gross errors during data storage/retrieval.
The recommendations in this document are primarily for guidance. It is recognised
that while the recommendations presented do form a valid approach to the
evaluation of measurement uncertainty for many purposes, other suitable
approaches may also be adopted – see references in Section 9. Especially the
EURACHEM/CITAC-Guide /2/ is useful in cases where sufficient previous data is
not available, and therefore the mathematical analytical approach according to
GUM /1/ with all different steps is to be used.
Basic knowledge in the use of quality control and statistics is required. In order to
make it possible for the reader to follow the calculations, some raw data is given in
appendices
2.2 Comment to customers
Previously, laboratories usually reported uncertainty as the standard deviation
calculated from data for an internal control sample. The measurement uncertainty
also taking into account method and laboratory bias and using a coverage factor of
2, can give uncertainty values which may be a factor of 2 to 5 times higher than
previously (Figure 1). However, this does not reflect a change in the performance
of the laboratory, just a much better estimation of the real variation between
laboratories. In Figure 1, the ammonium results from two laboratories are in good
agreement – the difference is about 5 %. You can see this if you look to the right
where measurement uncertainty is calculated correctly, but not if you look to the
left, where the uncertainty is calculated directly from a control sample and
presented as the standard deviation (± 1s).
Page 3 of 41
250
Measurement
uncertainty from...
240
Lab 1
230
Lab 2
Ammonium µg/l
220
210
200
190
180
Control sample
Rw
170
Control sample + CRM or
interlaboratory comparisons
160
150
Figure
1. Comparing ammonium results from two laboratories, Lab 1 = 199 µg/L and Lab6
0
2 = 188 µg/L. To the left the error bars are calculated from results on control samples
(± 1s) and to the right the error bars are expanded measurement uncertainty.
2.3 About Measurement Uncertainty
What is measurement uncertainty?
!
!
!
The number after ±
All measurements are affected by a certain error. The measurement uncertainty
tells us what size the measurement error might be. Therefore, the measurement
uncertainty is an important part of the reported result
Definition: Measurement uncertainty is ”A parameter associated with the result
of a measurement, that characterises the dispersion of the values that could
reasonably be attributed to the measurand” /1, 5/
Who needs measurement uncertainties?
!
!
The customer needs it together with the result to make a correct decision. The
uncertainty of the result is important, e.g. when looking at allowable (legal)
concentration limits
The laboratory to know its own quality of measurement and to improve to the
required quality
Why should the laboratory give measurement uncertainty?
!
!
As explained above, the customers need it to make correct decisions
An estimation of the measurement uncertainty is required in ISO 17025 /9/
Page 4 of 41
How is measurement uncertainty obtained?
!
!
!
The basis for the evaluation is a measurement and statistical approach, where
the different uncertainty sources are estimated and combined into a single
value
“Basis for the estimation of measurement uncertainty is the existing knowledge
(no special scientific research should be required from the laboratories).
Existing experimental data should be used (quality control charts, validation,
interlaboratory comparisons, CRM etc.)” /12/
Guidelines are given in GUM /1/, further developed in, e.g., EA guidelines
/12/, the Eurachem/Citac guide /2/, in a Eurolab technical report /3/ and in
ISO/DTS 21748 /8/
How is the result expressed with measurement uncertainty?
!
!
!
Measurement uncertainty should normally be expressed as U, the combined
expanded measurement uncertainty, using a coverage factor k = 2, providing a
level of confidence of approximately 95 %
It is often useful to state how the measurement uncertainty was obtained
Example, where ± 7 is the measurement uncertainty:
Ammonium (NH4-N) = 148 ± 7 µg/L. The measurement uncertainty, 7 µg/L
(95 % confidence level, i.e. the coverage factor k=2) is estimated from control
samples and from regular interlaboratory comparisons
How should measurement uncertainty be used?
!
!
It can be used as in Figure 1, to decide whether there is a difference between
results from different laboratories, or results from the same laboratory at
different occasions (time trends etc.)
It is necessary when comparing results to allowable values, e.g. tolerance limits
or allowable (legal) concentrations
Page 5 of 41
3 Calculation of expanded uncertainty, U - overview
A common way of presenting the different contributions to the total measurement
uncertainty is to use a so-called fish-bone (or cause-and-effect) diagram. We
propose a model (Figure 2), where either the reproducibility within-laboratory (Rw )
is combined with estimates of the method and laboratory bias, (error model in
Appendix 3) or the reproducibility sR is used more or less directly, ISO Guide
21748/8/. The alternative way is to construct a detailed fish-bone diagram and
calculate/estimate the individual uncertainty contributions. This approach may
prove very useful when studying or quantifying individual uncertainty components.
It has been shown, though, that in some cases this methodology underestimates the
measurement uncertainty /3/, partly because it is hard to include all possible
uncertainty contributions in such an approach. By using existing and
experimentally determined quality control (QC) and method validation data, the
probability of including all uncertainty contributions will be maximised.
Measurement uncertainty model – fish-bone diagram
covering the analytical process from sample arrival to report
QC - Reproducibility within
laboratory, Rw
(section 4)
Value
Analytical
Report
Method and lab bias
• Reference material
• Interlab comparison
• Validation
(section 5)
Decision maker
Customer
Reproducibility
between laboratories
sR
(section 6)
Figure 2. Measurement uncertainty model (fish-bone diagram), where the reproducibility
within-laboratory is combined with estimates of the method and laboratory bias.
Alternatively, according to ISO guide 21748 /8/, the combined uncertainty uc can be directly
estimated from the reproducibility between laboratories (sR). This approach is treated in
section 6
Page 6 of 41
3.1 Customer needs
Before calculating or estimating the measurement uncertainty, it is recommended
to find out what are the needs of the customers. After that, the main aim of the
actual uncertainty calculations will be to find out if the laboratory can fulfil the
customer demands with the analytical method in question. However, customers are
not used to specifying demands, so in many cases the demands have to be set in
dialogue with the customer. In cases where no demands have been established, a
guiding principle could be that the calculated expanded uncertainty, U, should be
approximately equal to, or less than, 2 times the reproducibility, sR.
3.2 Flow scheme for uncertainty calculations
The flow scheme presented in this section forms the basis for the method outlined
in this handbook. The flow scheme, involving 6 defined steps, should be followed
in all cases. The example with NH4-N in water shows the way forward for
calculating the measurement uncertainty using the flow scheme. Explanations of
the steps and their components will follow in the succeeding chapters. For each
step, there may be one or several options for finding the desired information.
Background for the NH4-N example – automatic photometric method: The
laboratory has participated in 6 interlaboratory comparisons recently. All results
have been somewhat higher than the nominal value. The laboratory therefore
concludes that there may be a small positive bias. On average, the bias has been
+2.2 %. This bias is considered small by the laboratory and is not corrected for in
their analytical results, but exists, and is thus another uncertainty component.
For this method, the main sources of uncertainty are contamination and variation in
sample handling, both causing random uncertainty components. These uncertainty
sources will be included in the calculations below.
Page 7 of 41
Step
Action
Specify Measurand
Example – Ammonium NH4-N
Ammonium is measured in water according to
EN/ISO 11732 /11/. The customer demand on
expanded uncertainty is ± 10 %
Quantify Rw comp.
A control sample
B possible steps not
covered by the
control sample
A: Control limits are set to ± 3.34 %
(95 % confidence limit)
3
Quantify bias comp.
From interlaboratory comparisons over the last 3
years the bias result were 2.4; 2.7; 1.9: 1.4; 1.8:
and 2.9. The root mean square (RMS) of the bias
is 2.25 %. The uncertainty of the nominal values
is u(Cref) = 1.5 %.
(see Appendix 4 for explanations)
4
Convert components
to standard
uncertainty u(x)
Confidence intervals and similar distributions
can be converted to standard uncertainty /1, 2,
14/.
u(Rw) = 3.34/2 = 1.67 %
1
2
B: The control sample includes all analytical
steps.
u(bias) = RMSbias + u(Cref )2
2
= 2.252 + 1.5 = 2.71 %
5
Calculate combined
standard uncertainty,
uc
Standard uncertainties can be summed by taking
the square root of the sum of the squares
uc = u(Rw ) 2 + (u(bias)) = 1.672 + 2.712 = 3.18
2
6
Calculate expanded
uncertainty,
U = 2 ⋅ uc
The reason for calculating the expanded
uncertainty is to reach a high enough confidence
(app. 95 %) in that the “true value” lies within
the interval given by the measurement result ±
the uncertainty. U = 2 ⋅ 3.18 = 6.36 ≈ 6 %.
The measurement uncertainty for NH4-N will thus be reported as ± 6 % at this
concentration level.
Page 8 of 41
3.3 Summary table for uncertainty calculations
The results of the calculations done in the flow scheme will then be summarised in
a summary table.
Ammonium in water by EN/ISO 11732
Measurement uncertainty U (95 % confidence interval) is estimated to ± 6 %. The
customer demand is ± 10 %. The calculations are based on control chart limits and
interlaboratory comparisons.
Value
Relative
u(x)
Comments
Reproducibility within-laboratory, Rw
Control sample
X
Rw
= 200 µg/L
Other components
Control limits is set
to ± 3.34 %
1.67 %
--
Method and laboratory bias
Reference material
bias
--
Interlaboratory
comparisons
bias
RMSbias=
Recovery test
bias
2.25 %
u(Cref) = 1.5 %
2.71 %
u(bias) =
RMSbias + u(Cref )2
2
--
Reproducibility between laboratories
Interlaboratory
comparisons
R
--
Standard method
R
--
8.8 %
Data - see Section
6.2
Combined uncertainty, uc is calculated from the control sample limits and bias
estimation from interlaboratory comparisons. The sR from interlaboratory comparisons can also be used (see 6.2) if a higher uncertainty estimation is acceptable.
Measurand
Ammonium
Combined Uncertainty uc
1.67 2 + 2.67 2 = 3.18 %
Expanded Uncertainty U
3.18 · 2 = 6.4 ≈ 6 %
Page 9 of 41
4 Reproducibility within-laboratory - u(Rw)
In this section the most common ways of estimating the reproducibility withinlaboratory component, u(Rw), for the measurement uncertainty calculation are
explained:
•
Stable control samples covering the whole analytical process. Normally
one sample at low concentration level and one at a high concentration
level.
•
Control samples not covering the whole analytical process. Uncertainties
estimated from control samples and from duplicate analyses of real
samples with varying concentration levels.
•
Unstable control samples.
It is of utmost importance that the estimation must cover all steps in the analytical
chain and all types of matrices – worst-case scenario. The control sample data
should be run in exactly the same way as the samples e.g. if the mean of duplicate
samples is used for ordinary samples, then the mean of duplicate control samples
should be used for the calculations.
It is likewise important to cover long-term variations of some systematic
uncertainty components within the laboratory, e.g. caused by different stock
solutions, new batches of critical reagents, recalibrations of equipment, etc. In
order to have a representative basis for the uncertainty calculations and to reflect
any such variation the number of results should ideally be more than 50 and cover
a time period of approximately one year, but the need differs from method to
method.
4.1 Customer demands
Some laboratories choose to use the customer demand when setting the limits in
their control charts. The actual performance of the method is not interesting, as
long as it meets the customer demands on expanded uncertainty. If, for example,
the customer asks for data with an (expanded) measurement uncertainty of ± 10 %,
then, from our experience, a good starting point is to set the control limits ± 5 %.
The u(Rw) used in the calculations will then be 2.5 %.1 This is just a proposal and
the measurement uncertainty calculations will show if these control limits are
appropriate.
4.2 Control sample covering the whole analytical process
When a stable control sample is covering the whole analytical process and has a
matrix similar to the samples, the within-laboratory reproducibility at that
concentration level can simply be estimated from the analyses of the control
1
Treating the control limits according to GUM /1/ as type B estimate with 95 %
confidence limit
Page 10 of 41
samples. If the analyses performed cover a wide range of concentration levels, also
control samples of other concentration levels should be used. Example: For NH4-N
two control sample levels (20 µg/L and 250 µg/L) were used during year 2002. The
results for the manual analysis method are presented in the table below.
Value
Relative u(x)
Comments
Reproducibility within-laboratory, Rw
Control sample 1
X = 20.01 µg/L
sRw
Control sample 2
X = 250.3 µg/L
sRw
Standard
deviation 0.5
µg/L
Standard
deviation 3.7
µg/L
--
Other components
2.5 %
1.5 %
From
measurements in
2002, n = 75
From
measurements in
2002, n = 50
4.3 Control sample for different matrices and concentration
levels
When a synthetic control solution is used for quality control, and the matrix type of
the control sample is not similar to the natural samples, we have to take into
consideration uncertainties arising from different matrices. Example: To estimate
the matrix based uncertainties, duplicate analysis of ammonium is performed, and
the sr is estimated from the corresponding R%-chart (Range%-chart /13/), giving
the repeatability of analysing natural samples having a normal matrix variation at
different concentration levels.
The data set consists of 73 duplicate analyses in the range of 2 µg/L – 16000 µg/L.
Most of the results were below 200 µg/L. The data is divided into two parts:
< 15 µg/L and > 15 µg/L
The sr can be estimated from R%-charts constructed for both concentration ranges.
The data is given in Appendix 5. The standard deviation is estimated from the
range (see Appendix 8): s = range / 1.128 .
Value
Relative u(x)
Comments
Reproducibility within-laboratory, Rw
Variation from
duplicate analysis
2-15 µg/L:
> 15 µg/L:
sR
5.7 %
3.6 %
n = 43 ( X = 6.50
µg/L)
n = 30 ( X = 816
µg/L)
Page 11 of 41
At low levels it is often better to use an absolute uncertainty rather than a relative,
as relative numbers tend to become extreme at very low concentrations. In this
example the absolute u(r) becomes 0.37 µg/L for the natural sample (mean
concentration 7 µg/L) and 0.5 µg/L for the control sample in Section 4.2 (mean
concentration 20 µg/L).
As the estimate from duplicate analysis gives the repeatability component (sr) only,
it should be combined with the control sample results from Section 4.2 to give a
better estimate of sRw. This way, the repeatability component will be included two
times, but it is normally small in comparison to the between-days variation.
Value
u(x)
Comments
Reproducibility within-laboratory, Rw
Low level
(2-15 µg/L)
High level
(> 15 µg/L)
sRw
sRw
0.5 µg/L from 0.6 µg/L
control sample
and 0.37 µg/L
from duplicates
1.5% from
3.9 %
control sample
and 3.6% from
duplicates
Absolute
u(x) =
0.5 2 + 0.37 2
Relative
u(x) =
1.5 2 + 3.6 2
It can be noticed that the sample matrix has some effect on the variation of the
results. The reason for this is not only the matrix, but also the concentration level,
as almost all of the duplicate analyses were performed at a concentration level
below 10 µg/L. The quantification limit of the measurement was 2 µg/L and the
relative standard deviation usually becomes higher near that limit (cf. Figures 4 and
5 in Section 6.3).
4.4 Unstable control samples
If the laboratory does not have access to stable control samples, the reproducibility
can be estimated using analysis of natural duplicate samples. The results from the
duplicate sample analysis can either be treated in an R-chart, where the difference
between the first and second analysis is plotted directly, or as an R %-chart, where
the absolute difference between the sample pair is calculated in % of the average
value of the sample pair. The latter approach is particularly useful when the
concentration varies a lot from time to time.
In this example, duplicate samples for oxygen have been analysed on 50 occasions.
The raw data is given in Appendix 6. The concentration variation is limited, so an
R-chart approach is chosen. The difference between the first and the second
analysis is calculated and plotted in a chart, see Figure 3. In this case, the second
result is always subtracted from the first when constructing the R-chart, as it is
important to look for systematic differences between the first and the second
Page 12 of 41
sample. The standard deviation for the results can be estimated from the average
range of the duplicate samples (see Appendix 8), and in this case becomes 0.024.
The control limits at ±2s thus lies at ±0.048. The average value of the first
determination is 7.53, and the sr thus equals 100·0.024/7.53 = 0.32 %.
Oxygen in seawater, double samples
Difference, mg/L
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
0
10
20
30
40
50
Analysis no.
Figure 3. The difference between oxygen duplicate measurements plotted in
an R-chart
However, this only gives the within-day variation (repeatability, sr) for sampling
and measurement, and there will also be a “long-term” uncertainty component from
the variation in the calibration (here the thiosulphate used for titrating or the
calibration of the oxygen probe, depending on method). For this particular analysis,
the uncertainty component from the long-term variation in calibration is hard to
measure, as no stable reference material or CRM is available for dissolved oxygen.
One method would be to calibrate the same thiosulphate solution several times
during a few days time, and use the variation between the results. Here we choose
to estimate that component by a qualified guess, but laboratories are encouraged to
also try the experimental approach.
The total reproducibility within-laboratory for dissolved oxygen thus becomes:
Page 13 of 41
Value
Relative u(x)
Comments
Reproducibility within-laboratory, Rw
Duplicate
measurements of
natural samples,
difference used in rchart
Estimated variation
from differences in
calibration over time
sR
s = 0.024 mg/L 0.32 %
X = 7.53 mg/L
s = 0.5 %
0.5 %
Combined uncertainty for Rw
Repeatability +
reproducibility in
calibration
Page 14 of 41
0.32 2 + 0.5 2 = 0.59 %
Measurements in
2000-2002,
n= 50
Estimate, based
on experience
5 Method and Laboratory bias – u(bias)
In this chapter the most common ways of estimating the bias components will be
outlined, namely the use of CRM, participation in interlaboratory comparisons
(proficiency test) and recovery tests. Sources of bias should always be eliminated if
possible. According to GUM /1/ a measurement result should always be corrected
if the bias is significant and based on reliable data such as a CRM. However, even
if the bias is zero, it has to be estimated and treated as an uncertainty component. In
many cases the bias can vary depending on changes in matrix. This can be reflected
when analysing several matrix CRMs, e.g. the bias could be both positive and
negative. Examples are given and explained for the proposed calculations.
For every estimation of the uncertainty from the method and laboratory bias, two
components have to be estimated to obtain u(bias):
1) the bias (as % difference from the nominal or certified value)
2) the uncertainty of the nominal/certified value, u(Cref) or u(Crecovery)
The uncertainty of the bias, u(bias) can be estimated by
u (bias ) = RMSbias + u (Cref ) where RMSbias =
2
2
∑ (bias )
2
i
n
and if only one CRM is used also the sbias have to be included and u(bias) can the
be estimated /14, 15/2 by
2
s 
u (bias ) = (bias ) +  bias  + u (Cref ) 2
 n
2
5.1 Certified Reference Material
Regular analysis of a CRM can be used to estimate the bias. The reference material
should be analysed in at least 5 different analytical series (e.g. on 5 different days)
before the values are used.
In this example the certified value is 11.5 ± 0.5, with a 95 % confidence interval.
Uncertainty component from the uncertainty of the certified value
Step
Step
Convert the confidence
The confidence interval is ± 0.5. Divide this by 1.96
interval to u(Cref)
to convert it to standard uncertainty:
0.5/1.96 = 0.26
Convert to relative
100·(0.26/11.5) = 2.21%
uncertainty u(Cref)
Page 15 of 41
3
4
Quantify Method and
laboratory bias
Convert components
to standard
uncertainty u (x)
bias = 100 ⋅ (11.9 - 11.5)/11.5 = 3.48 %
sbias = 2.2 % (n = 12)
u(Cref) = 2.21 %
u (bias ) =
(bias )2 +  sbias 
2
 n
+ u (Cref ) 2 =
2
(3.48) +  2.2  + 2.212 = 4.2 %
 12 
2
If several CRM:s are used, we will get different values for the bias. The
uncertainty of the bias estimation will be calculated in the following way (see also
section 5.2).
3
Quantify Method and
laboratory bias
bias CRM1 is 3.48%, s=2,2 (n=12), u(Cref)=2.21 %
bias CRM2 is -0.9% s=2,0 (n=7)), u(Cref)=1.8 %
bias CRM3 is 2.4%, s= 2,8 (n=10), u(Cref)=1.8 %
For the bias the RMSbias = 2.50
mean u(Cref)=1,9 %
4
Convert components
to standard
uncertainty u (x)
u (bias ) = RMS bias + u (Cref ) 2
Page 16 of 41
2
2.50 2 + 1.92 = 3.1%
5.2 Interlaboratory comparisons
In this case the results from interlaboratory comparisons are used in the same way
as a reference material, i.e. to estimate the bias. In order to have a reasonably clear
picture of the bias from interlaboratory comparison results, a laboratory should
participate at least 6 times within a reasonable time interval.
Biases can be both positive and negative. Even if the results appear to give positive
biases on certain occasions and negative on others, all bias values can be used to
estimate the uncertainty component, RMSbias.
The way forward is very similar to that for reference materials. However, the
estimation of the bias from interlaboratory comparisons has more uncertainty to it,
and thus usually becomes a bit higher than if CRMs are used. This is partly due to
the fact that the certified value of a CRM normally is better defined than a nominal
or assigned value in an interlaboratory comparison exercise. In some cases the
calculated uncertainty u(Cref) from an interlaboratory comparison becomes too
high and is not valid for estimating the u(bias).
Uncertainty component from the uncertainty of the nominal value
Step
Example
Find the between laboratory The sR has been on average 9% in the 6 exercises.
standard deviations, sR, for
the exercises.
Calculate u(Cref)
Mean number of participants = 12.
u (Cref ) =
sR
9
=
= 2.6 %
n
12
The bias has been 2 %, 7 %, -2 %, 3 %, 6 % and 5%, in the 6 interlaboratory
comparisons where the laboratory has participated.
3
Quantify Method and
laboratory bias
4
Convert components
to standard
uncertainty u (x)
= 4.6 %,
u(Cref)= 2.6 %
RMSbias
u(bias) = RMSbias + u(Cref ) 2 =
2
= 4.62 + 2.62 = 5.3%
Page 17 of 41
5.3 Recovery
Recovery tests, for example the recovery of a standard addition to a sample in the
validation process, can be used to estimate the systematic error. In this way,
validation data can provide a valuable input to the estimation of the uncertainty.
In an experiment the recoveries for an added spike were 95 %, 98 %, 97 %, 96 %,
99 % and 96 % for 6 different sample matrices. The average is 96.8 %. The spike
of 0.5 mL was added with a micropipette.
Uncertainty component from the definition of 100% recovery, u(Crecovery)
Step
Example
Uncertainty of the 100%
u(conc) - Certificate ± 1.2 % (95 % conf. limit) gives =
recovery. Main components 0.6 %
are concentration, u(conc) u(vol) - This value can normally be found in the
of standard and volume
manufacturer’s specifications, or better use the limits
added u(vol)
specified in your laboratory. Max bias 1 %
(rectangular interval) and repeatability max 0.5 %
2
 1 
u (vol ) = 
 + 0.5 2 = 0.76 %
 3
Calculate u(Crecovery)
u (conc) 2 + u (vol ) 2 = 0.6 2 + 0.76 2 = 1.0 %
= 3.44 %
u(Crecovery)=1.0 %
3
Quantify Method and
laboratory bias
RMSbias
4
Convert components
to standard
uncertainty u (x)
u (bias ) = RMS bias + u (Cref ) 2 =
Page 18 of 41
2
= 3.44 2 + 1.0 2 = 3.6 %
6 Reproducibility between laboratories, sR
If the demand on uncertainty is low, it can be possible to directly use the sR from
interlaboratory comparisons as an approximation of uc /8/. In such case the
expanded uncertainty becomes U = 2 ⋅ s R . This may be an overestimate depending
on the quality of the laboratory – worst-case scenario. It may also be an
underestimate due to sample inhomogenity or matrix variations.
6.1 Data given in standard method
In order to use a figure taken directly from the standard method, the laboratory
must prove that they are able to perform in accordance with the standard method
/8/, i.e. demonstrating control of bias and verification of the repeatability, sr.
Reproducibility data can either be given as a standard deviation sR or as
reproducibility limit R and then sR = R/2.8
The example below is taken from ISO/DIS 15586 Water Quality — Determination
of trace elements by atomic absorption spectrometry with graphite furnace. The
matrix is wastewater. Combined uncertainty in wastewater, uc, is taken from the sR
from interlaboratory comparison exercises quoted in the ISO method.
Table 1 ISO/DIS 15586 - Results from the interlaboratory comparison – Cd in water with
graphite furnace AAS. The wastewater was digested by the participants.
n Outliers Nominal value Mean Recovery,
Cd
µg/L
µg/L
%
sr
sR
%
%
Synthetic
Lower
33
1
0.3
0.303
101
3.5 17.0
Synthetic
Higher 34
2
2.7
2.81
104
1.9 10.7
Fresh water Lower
31
2
0.572
2.9 14.9
Fresh water Higher 31
3
3.07
2.1 10.4
Waste
water
2
1.00
3.1 27.5
Measurand
Cd
27
Combined Uncertainty uc
uc = 27.5 %
Expanded Uncertainty U
2·uc = 55 % ≈ 50 %
6.2 Data from interlaboratory comparisons
Interlaboratory comparisons are valuable tools in uncertainty evaluation. The
reproducibility between laboratories is normally given directly in reports from the
exercises as sR.
These data may well be used by a laboratory (having performed satisfactorily in the
comparisons) as the standard uncertainty of the analysed parameter, provided that
the comparison covers all relevant uncertainty components and steps (see /9/,
Page 19 of 41
section 5.4.6.3). For example, a standard deviation in an interlaboratory
comparison, sR, can be directly used as a combined standard uncertainty, uc.
Table 2 Summary results (mean values) from 10 interlaboratory comparisons that Lab A
has participated in. The reproducibility standard deviation is given in absolute units, sR and
in relative units sR %.
Variable
pH
Conductivity,
mS/m
Alkalinity, mmol/L
Turbidity, FNU
NH4-N, µg/L
NO3-N, µg/L
Nominal
value
7.64
12.5
Lab A %
deviation
-0.037
-2.8
0.673
1.4
146
432
2.3
-9.1
2.2
-1.6
sR
sR
%
0.101
0.40 3.2
No. of
labs
90
86
5
6
0.026
0.1
12.0
16.3
60
44
34
39
3
3
5
6
(abs)
3.9
14.2
8.8
3.7
Excluded
In Table 2 we find that for conductivity, for instance, the mean value for the results
from 10 interlaboratory comparisons is 12.5 mS/m. The reproducibility relative
standard deviation is 0.4 (3.2 %), which is an average (or pooled) standard
deviation between the laboratories in the different interlaboratory comparisons and
this value can be taken as an estimate of combined uncertainty i.e.
uc (conductivity) = sR = 0.4 mS/m, thus U = 2·0.4 = 0.8 mS/m
If we take the ammonium results, we have a mean nominal value of 146 µg/L, and
we find that the reproducibility, sR, is 8.8 %. Thus U = 2·8.8 = 17.6 = 18 % at this
concentration level.
Comment: In Section 3 the expanded uncertainty for ammonium is 5 % using an
automated method in one highly qualified laboratory.
mS/m
Page 20 of 41
7 Examples
In this chapter, practical examples on how measurement uncertainty can be
calculated using the method of this handbook are given.
7.1 Ammonium in water
Ammonium in water has already been treated in section 3.2 and section 6.2 . The
results are summarised in Table 3.
Table 3 Measurement uncertainty of ammonium in water – comparison of different
calculations
Uncertainty calculations
based on
Control sample +
proficiency testing
Interlaboratory
comparisons
Relative expanded
uncertainty, U
±6%
± 18 %
Comment
Uncertainty for one good
laboratory- level 200 µg/L.
Uncertainty in general among
laboratories – level 150 µg/L
7.2 BOD in wastewater
Biological Oxygen Demand, BOD, is a standard parameter in the monitoring of
wastewater. This example shows how data from ordinary internal quality control
can be used together with CRM results or data from interlaboratory comparison
exercises to calculate the within-lab reproducibility and bias components of the
measurement uncertainty. The results are summarised in Table 4
Table 4 Measurement uncertainty of BOD in water - comparison of different
calculations
Uncertainty calculations
based on
Control sample + CRM
Control sample +
interlaboratory comparisons
Interlaboratory
comparisons
Relative expanded
uncertainty, U
± 10 %
± 10 %
± 16 %
Comment
n = 3, unreliable
estimate
Uncertainty in general
among laboratories
For BOD at high concentrations, using the dilution analytical method, the major
error sources are the actual oxygen measurement and variation in the quality of the
seeding solution. These errors will be included in the calculations.
The raw data from the internal quality control, using a CRM, used for the
calculations is shown in Appendix 7.
Page 21 of 41
The laboratory has only participated in three interlaboratory comparison exercises
the last 2 years (Table 5). At least six would be needed, so here we estimate the
bias two different ways – with CRM and with interlaboratory comparisons.
Table 5 BOD - results from interlaboratory comparisons
Exercise
1
2
3
X
RMSbias
3
Nominal
value
mg/L
154
219
176
Laboratory
result
mg/L
161
210
180
Bias
sR
Number of labs
%
+ 4.5
- 4.1
+2.3
+0.9
3.76
%
7.2
6.6
9.8
7.873
-
23
25
19
22.3
-
If sR or the number of participants vary substantially from exercise to exercise, then a
pooled standard deviation will be more correct to use. In this case, where the variation in sR
is limited, we simply calculate the mean sR (the corresponding pooled standard deviation
becomes 7.82, an insignificant difference).
Page 22 of 41
Example A: BOD with Internal quality control + a CRM
Step
1
2
Action
Specify Measurand
Example: BOD in wastewater
BOD in wastewater, measured with EN1899-1
(method with dilution, seeding and ATU). The
demand on uncertainty is ± 20 %.
Quantify u(Rw)
A: The control sample, which is a CRM, gives
an s = 2.6 % at a level of 206 mg/L O2. s = 2.6
% is also when setting the control chart limits.
A control sample
B possible steps not
covered by the
control sample
3
Quantify Method and
laboratory bias
B: The analysis of the control sample includes
all analytical steps after sampling
The CRM is certified to 206 ±5 mg/L O2. The
average result of the control chart is 214.8.
Thus, there is a bias of 8.8 mg/L = 4.3 %.
The sbias is 2.6 % (n=19)
The u(Cref) is 5 mg/L / 1.96 = 1.2 %
4
Convert components to
standard uncertainty
u(x)
u(Rw) = 2.6 %
2
u(bias) = bias2 +
sbias
+ u(Cref )2
n
2
 2.6 
= 4.32 + 
 + 1.22 = 4.5 %
 19 
5
6
Calculate combined
standard uncertainty, uc
Calculate expanded
uncertainty, U = 2 ⋅ u c
uc =
2.6 2 + 4.5 2 = 5.2 %
U = 2 ⋅ 5.2 = 10.4 ≈ 10 %
Page 23 of 41
Example B: BOD with Internal quality control + interlaboratory comparison
results
Step
1
2
Action
Specify Measurand
Example: BOD in wastewater
BOD in wastewater, measured with EN1899-1
(method with dilution, seeding and ATU). The
demand on uncertainty is ± 20 %.
Quantify u(Rw)
A: The control sample, which is a CRM, gives
an s of 2.6 % at a level of 206 mg/L O2. s = 2.6
% is also used as s when setting the control
chart limits.
A control sample
B possible steps not
covered by the
control sample
3
4
B: The analysis of the control sample includes
all analytical steps after sampling
Quantify Method and
laboratory bias
Data from Table 5
RMSbias
= 3.76
Convert components to
standard uncertainty
u(x)
u(Rw) = 2.6 %
u (Cref ) =
sR
7.9
=
= 1.67
n
22.3
u(bias) = RMSbias + u(Cref)2 =
2
3.762 +1.672 = 4.11%
5
6
Calculate combined
standard uncertainty, uc
Calculate expanded
uncertainty, U = 2 ⋅ u c
Page 24 of 41
uc =
2.6 2 + 4.112 = 4.86 %
U = 2 ⋅ 4.86 = 9.7 ≈ 10 %
7.3 PCB in sediment
In this example, the u(Rw) is estimated from a quality control sample and the
u(bias) is estimated from two different sources: in the first example the use of a
CRM and in the second example participation in interlaboratory comparisons. In
the summary table both ways of calculating the u(bias) will be compared.
For this analysis, the sample-work up is a major error source (both for random and
systematic errors), and it is thus crucial that this step is included in the calculations.
The number of interlaboratory comparisons is too few to get a good estimate.
Example C: PCB with Internal quality control + a CRM
Step
1
2
Action
Specify Measurand
Example: PCB in sediment
Sum of 7 PCB:s in sediment by extraction and
GC-MS(SIM). Demand on expanded
uncertainty is ± 20 %.
Quantify u(Rw)
A: The control sample, which is a CRM, gives
an sRw = 8 % at a level of 150 µg/kg dry matter.
sRw = 8 % is also used when setting the control
chart limits.
B: The analysis of the control sample includes
all steps except for drying the sample to
determine the dry weight. The uncertainty
contribution from that step is considered small
and is not accounted for.
A control sample
B possible steps not
covered by the
control sample
3
Quantify
method and
laboratory bias
The CRM is certified to 152 ± 14 µg/kg. The
average result of the control chart is 144. Thus,
there is a bias = 5.3 %.
The sbias = 8 % (n=22)
u(Cref) 14 µg/kg/1.96, which is 4.7 % relative.
4
Convert components to
standard uncertainty
u(x)
u(Rw) = 8 %
2
u(bias) = bias2 +
sbias
+ u(Cref )2
n
2
 8 
= 5.32 + 
 + 4.72 = 7.29
 22 
5
Calculate combined
standard uncertainty, uc
uc = 8 2 + 7.29 2 = 10.8 %
6
Calculate expanded
uncertainty, U = 2 ⋅ u c
U = 2 ⋅ 10.8 = 21.6 ≈ 22 %
Page 25 of 41
Example D: PCB with Internal quality control + interlaboratory comparison
Step
1
2
Action
Specify Measurand
Example: PCB in sediment
Sum of 7 PCB:s in sediment by extraction and
GC-MS(SIM). Demand on expanded
uncertainty is 20 %.
Quantify u(Rw)
A: The control sample, which is a stable inhouse material, gives sRw = 8 % at a level of
150 µg/kg dry matter. sRw = 8 % is also used
as s when setting the control chart limits.
B: The analysis of the control sample includes
all steps except for drying the sample to
determine the dry weight. The uncertainty
contribution from that step is considered small
and is not accounted for.
A control sample
B possible steps not
covered by the
control sample
3
Quantify Method and
laboratory bias
Participation in 3 interlaboratory comparisons
with concentration levels similar to the
internal quality control. The bias in the 3
exercises has been –2 %, -12 % and –5 %.
RMSbias = 7.6
The sR in the three exercises has been 12 %,
10 % and 11 %, on average sR = 11 % (n=14)
u (Cref ) =
4
Convert components to
standard uncertainty
u(x)
11
14
= 2.9 %
The u(Rw) is 8 %
u(bias) = RMSbias + u(Cref)2 =
2
7.62 + 2.92 = 8.1%
5
Calculate combined
standard uncertainty, uc
6
Calculate expanded
uncertainty, U = 2 ⋅ u c
Page 26 of 41
uc = 8 2 + 8.12 = 11.4
U = 2 ⋅11.4 = 22.8 ≈ 23 %
Summary table for PCB measurement uncertainty
calculations
PCB in sediment by extraction and GC-MS (SIM)
Measurement uncertainty U (95 % confidence interval) is estimated to ± 20 %
(relative) for 7 PCB:s in sediments at a level of 150 µg/kg dry weight. The
customer demand is ± 20 %. The calculations are based on internal quality control
using a stable sample, CRM and the participation in a limited amount of
interlaboratory comparison exercises.
Value
u(x)
Comments
Reproducibility within-laboratory, Rw
Control sample
X = 160 µg/kg
dry weight
Other components
u(Rw)
12.8 µg/kg dry
weight
8%
too small to be considered
Method and laboratory bias
Bias: 5.3 %
Reference
material
u(bias) = 5.88 u(bias)=
sbias = 8 ; n = 22
Interlaboratory
comparison
n=3
u(Cref) = 4,7 %
RMSbias = 7.6
u(Cref) = 2.9 %
bias 2 +
u(bias) = 8.1
s bias
n
2
+ u(Cref ) 2
u(bias)=
RMS bias + u (Cref ) 2
2
Combined uncertainty, uc, is calculated from internal quality control and the
maximum bias - interlaboratory comparisons.
Measurand
PCB
Combined Uncertainty uc
uc =
8 2 + 8.12 = 11.4
Expanded Uncertainty U
U = 2 ⋅ u c = 2 ⋅ 11.4 = 22.8 ≈ 23 %
Conclusion: In this case the calculation of the u(bias) gives similar results
regardless of whether CRM or interlaboratory comparison results are used.
Sometimes interlaboratory comparisons will give considerably higher values, and it
might in such cases be more correct to use the CRM results.
Page 27 of 41
7.4 Concentration ranges
The measurement uncertainty will normally vary with concentration, both in
absolute and relative terms. If the concentration range of the reported data is large,
it is thus often necessary to take this into account. For lead (Pb) in water, a
recovery experiment was carried out a number of times to investigate within-lab
reproducibility over the measurable range – the major component of the
measurement uncertainty at low levels. The following results were obtained:
Table 6 Within-lab reproducibility and recovery for Pb determined with ICP-MS at
different concentration levels.
Addition,
µg/L
0.01
0.1
0.4
1
10
10
100
Pb, %
recovery
109.7
125.2
91.8
98.4
98
100.5
105.5
s, %
53.8
12.1
5
3.0
1.7
1.3
1.4
60
50
s, %
40
30
20
10
0
0.01
0.1
1
10
100
Concentration (log scale)
Figure 4 Within-lab reproducibility for Pb over the concentration range
It is clear from the results that the measurement uncertainty, here represented by s,
is strongly concentration dependent. Two approaches are recommended for using
these data:
(1) To divide the measurable range into several parts, and use a fixed relative
measurement uncertainty or absolute uncertainty – see Table 7.
Page 28 of 41
Table 7 Within-lab reproducibility for Pb divided into three concentration ranges
Within-lab reproducibility Pb
Range (µg/L)
s(rel)
s(rel) or (abs)
< 0.1
50 %
0.01 (µg/L)
0.1 - 10
10 %
10 %
> 10
2%
2%
In the second column s is relative and given in %. In the third column s is also
relative but an absolute value is given in the lower range close to the detection
limit.
(2) To use an equation that describes how the measurement uncertainty varies with
concentration
Plotting s % against 1/concentration gives a straight line, and a relatively simple
equation. (see Figure 5).
15
s 10
%
5
y = 1.06x + 1,77
2
R = 0.9798
0
0
2
4
6
8
10
12
1/concentration
Figure 5: The relationship between within-lab reproducibility and the inverted
concentration for Pb in the range 0.1 – 100 µg/L.
The straight-line equation above tells us that the within-lab reproducibility equals
1.06 multiplied with 1/concentration plus 1.77. For example, at a concentration of 2
µg/L the within-lab reproducibility becomes 1.06·1/2 + 1.77 = 2.3 %. When
reporting to customers, the laboratory can choose between quoting the formula or
calculating the measurement uncertainty for each value, using the formula. For
further reading, see for example /2/.
Page 29 of 41
8 Reporting uncertainty
This is an example on what a data report could look like, when measurement
uncertainty has been calculated and is reported together with the data. The
company and accreditation body logotypes are omitted, and the report does not
contain all information normally required for an accredited laboratory. It is
recommended to use either relative or absolute values for the benefit of the
customer.
Analytical Report
Sample identification: P1 – P4
Samples received: 14 December 2002
Analysis period: 14 –16 December 2002
Results
NH4-N (µg/L):
Sample
P1
P2
P3
P4
Result
103
122
12
14
U
±6%
±6%
±10%
±10%
Method
23B
23B
23B
23B
TOC (mg/L)
Sample
P1
P2
P3
P4
Result
40
35
10
9
U
±4.0
±3.5
±1.0
±0.9
Method
12-3
12-3
12-3
12-3
Signed: Dr Analyst
Page 30 of 41
The laboratory should also prepare a note explaining how the measurement
uncertainty has been calculated for the different parameters. Normally, such an
explanatory note should be communicated to regular customers and other
customers who ask for information. An example is given below:
Note on measurement uncertainty from Dr Analyst’s laboratory
Measurement uncertainty:
U = expanded Measurement Uncertainty, estimated from control sample
results, interlaboratory comparison and the analyses of CRMs, using a
coverage factor of 2 to reach approximately 95% confidence level.
NH4-N: U is estimated to 6% above 100 µg/L and 10% below 100 µg/L.
TOC: U is estimated to 10% over the whole concentration range.
References:
• Guide To The Expression Of Uncertainty In Measurement (GUM)
• Quantifying Uncertainty in Analytical Measurement.
EURACHEM/CITAC Guide
• Handbook for calculation of measurement uncertainty in
environmental laboratories
Page 31 of 41
9 References
1. Guide To The Expression Of Uncertainty In Measurement (GUM). BIPM,
IEC, IFCC, ISO, IUPAC, IUPAP, OIML. International Organization of
Standardization, Geneva Switzerland, 1st Edition 1993, Corrected and reprinted
1995.
2. Quantifying Uncertainty in Analytical Measurement. EURACHEM/CITAC
Guide, 2nd Edition, 2000
3. Measurement Uncertainty in Testing, Eurolab Technical Report No. 1/2002
4. Interlaboratory comparison test data, personal communication, H. Hovind,
NIVA, Norway.
5. International Vocabulary of Basic and General Terms in Metrology (VIM).
ISO, 1993
6. ISO/IEC 3534-1-2, Statistics – Vocabulary and symbols Parts 1-2
7. ISO 5725-1-6:1994, Accuracy (trueness and precision) of measurement
methods and results
8. ISO/DTS 21748:2003, Guide to the use of repeatability, reproducibility and
trueness estimates in measurement uncertainty estimation
9. EN ISO/IEC 17025:2000, General Requirements for the Competence of
Calibration and Testing Laboratories
10. ISO/TR 13530:1997, Water quality – Guide to analytical quality control for
water analysis
11. EN ISO 11732:1997, Water quality -- Determination of ammonium nitrogen by
flow analysis (CFA and FIA) and spectrometric detection
12. EA-4/xx: Proposed “EA guideline on The Expression of uncertainty in
quantitative testing”. Project to be finalised in 2003. (www.europeanaccreditation.org/)
13. ISO8258, First edition, 1991-12-15, Shewhart Control Charts
14. V. J. Barwick, Ellison L.R., Analyst, 1999, 124, 981-990
15. E. Hund, D.L. Massart and J. Smeyers-Verbeke, Operational definitions of
uncertainty. TrAC, 20 (8), 2001
Page 32 of 41
10 Appendices
Appendix 1: Empty flow scheme for calculations
Before starting: Always identify the main error sources, to make sure that they are
included in the calculations.
Step
1
2
Action
Specify Measurand
Measurand:
(measurand) in (matrix) by (method) The
customer demand on expanded uncertainty is
± _ %.
Quantify u(Rw)
A control sample
B possible steps not
covered by the control
sample
A:
3
Quantify method and
laboratory bias
4
Convert components to
standard uncertainty u(x)
5
Calculate combined
standard uncertainty, uc
6
Calculate expanded
uncertainty, U = 2 ⋅ u c
B:
Page 33 of 41
Appendix 2: Empty summary table
(measurand) in (matrix) by (method)
Measurement uncertainty U (95 % confidence interval) is estimated to ± _ %
(relative) for (measurand) in (matrix) at a level of _ (unit). The customer demand is
± _ %. The calculations are based on (control samples/control
limits/CRM/interlaboratory comparison/other).
Value
Relative u(x)
Comments
Reproducibility within-laboratory, Rw
Control sample
X
sRw
= (conc) (unit)
Other components
Method and laboratory bias
Reference material
bias
Interlaboratory
comparison
bias
Recovery test
bias
Reproducibility between laboratories
Interlaboratory
comparison
sR
Standard method
sR
Combined uncertainty, uc, is calculated from __ and bias from __.
Measurand
Combined Uncertainty uc
Expanded Uncertainty U
2 ⋅ uc =
Page 34 of 41
Appendix 3: Error model used in this handbook
This model is a simplification of the model presented in the ISO guide /8/:
y = m + (δ + B) + e
y
measurement result of a sample
m
expected value for y
δ
method bias
B
laboratory bias – the uncertainty for these are combined to u(bias)
e
random error at within-laboratory reproducibility conditions, Rw
Uncertainty estimation in section 3 to 5
u ( y ) 2 = s Rw + u (bias ) 2
2
s Rw
2
The estimated variance of e under within-laboratory reproducibility
conditions – intermediate precision. In the ISO guide the repeatability,
sr is used as an estimate of e.
u (bias ) 2 The estimated variance of method bias and laboratory bias.
Uncertainty estimation in section 6
The combined uncertainty u(y) or uc can also be estimated by from reproducibility
data.
u ( y ) 2 = s L + s r = s R - equation A6 ref. /8/
2
2
2
where sR2 is the estimated variance under reproducibility conditions and where sL2
is either the estimated variance of B if one method is used by all laboratories or an
estimated variance of B and δ if several different methods have been used in the
collaborative study and sr2 is the estimated variance of e.
Comment
For samples that are more inhomogeneous and have big variations in matrix the
estimation of the measurement uncertainty of the method can become too low.
However we recommend the use of repeatability limit for duplicate analyses
r = 2.8 ⋅ s r in order to control sample inhomogenity.
Page 35 of 41
Appendix 4: Uncertainty of bias for NH4-N in section 3.2
Results for a laboratory from interlaboratory comparisons of NH4-N in water.
Exercise
Nominal
value xref
mg/L
1999 1
2
2000 1
2
2001 1
2
Laboratory
result xi
mg/L
81
73
264
210
110
140
83
75
269
213
112
144
X
RMS
RMS
of the bias =
u(Cref) =
sR
n
=
∑ bias
n
2
i
=
Bias
sR
%
%
2.4
2.7
1.9
1.4
1.8
2.9
+ 2.18
2.25
10
7
8
10
7
11
8.8
-
Number
of labs
31
36
32
35
36
34
34
-
2.4 2 + 2.7 2 + ...2.9 2
= 2.25 % (rel)
n
8.8
= 1.5 % (rel)
34
A t-test shows that the bias (+2.18 %) is not significant (t = 0.6). However, in order
not to complicate the calculations when the bias is small, t-test are normally not
performed.
The mean value of sR is used. If differences in number of laboratories and sR are
very big pooled standard deviations should be used. In this case the pooled
standard deviation is 8.9 % for sR which is the same as the mean value of 8.8 %.
Page 36 of 41
Appendix 5: Raw data for NH4-N in section 4.3
The estimation of the standard deviation from the range is explained in Appendix 8
concentration < 15 µg/L
Sam ple
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
X1
7.46
9.01
3.6
6.48
14.49
10.84
4.61
2.6
2.8
5.84
2.12
2.3
2.52
3.71
7.43
8.83
9.12
8.24
2.62
3.33
2.69
12.09
4.24
10.49
3.68
9.37
2.22
6.1
2.96
14.02
4.24
5.1
2.78
8.52
12.82
3.17
11.28
14.31
4.01
3.27
9.98
12.56
3.35
X2
7.25
9.17
3.1
6.48
14.12
9.89
5
2.42
2.62
6.19
2.5
2.11
2.89
3.71
7.43
8.51
8.79
7.9
2.78
3.33
2.69
12.09
4.24
10.64
3.52
9.37
2.06
6.1
2.86
13.7
3.62
4.61
2.62
6.81
14.05
2.4
11.43
13.82
4.48
3.58
10.29
13.66
2.88
M ean:
7.355
9.090
3.350
6.480
14.305
10.365
4.805
2.510
2.710
6.015
2.310
2.205
2.705
3.710
7.430
8.670
8.955
8.070
2.700
3.330
2.690
12.090
4.240
10.565
3.600
9.370
2.140
6.100
2.910
13.860
3.930
4.855
2.700
7.665
13.435
2.785
11.355
14.065
4.245
3.425
10.135
13.110
3.115
6.499
0.210
-0.160
0.500
0.000
0.370
0.950
-0.390
0.180
0.180
-0.350
-0.380
0.190
-0.370
0.000
0.000
0.320
0.330
0.340
-0.160
0.000
0.000
0.000
0.000
-0.150
0.160
0.000
0.160
0.000
0.100
0.320
0.620
0.490
0.160
1.710
-1.230
0.770
-0.150
0.490
-0.470
-0.310
-0.310
-1.100
0.470
s(r) % = range(mean)/1.128 =
2.855
1.760
14.925
0.000
2.587
9.165
8.117
7.171
6.642
5.819
16.450
8.617
13.678
0.000
0.000
3.691
3.685
4.213
5.926
0.000
0.000
0.000
0.000
1.420
4.444
0.000
7.477
0.000
3.436
2.309
15.776
10.093
5.926
22.309
9.155
27.648
1.321
3.484
11.072
9.051
3.059
8.391
15.088
6.4363
5.71
= m ean range (% )
%
Page 37 of 41
concentration > 15 µg/L
X1
Sample
1 37.62
2 16.18
3 28.82
4 4490
5 135.7
6 62.56
7 158.9
8 16540
9 31.26
10 58.49
11 740.5
12 130.3
13 29.35
14 1372
15 36.55
16 22.57
17 34.75
18 92.93
19 40.6
20 80.36
21 15.76
22 78.22
23 48.89
24 17.65
25 36.56
26 51.89
27 197.5
28 70.32
29 29.99
30 31.9
X2
36.85
37.235
16.56
16.370
28.65
28.735
4413
4451.500
124.7
130.200
62.25
62.405
159.2
159.050
16080
16310.000
30.12
30.690
60.11
59.300
796.2
768.350
126.9
128.600
29.19
29.270
1388
1380.000
44.74
40.645
23.37
22.970
33.15
33.950
94.01
93.470
42.23
41.415
86.36
83.360
18.54
17.150
73.76
75.990
50.91
49.900
16.72
17.185
35.3
35.930
52.2
52.045
206.5
202.000
69.22
69.770
30.62
30.305
32.36
32.130
Mean: 816.331
0.770
-0.380
0.170
77.000
11.000
0.310
-0.300
460.000
1.140
-1.620
-55.700
3.400
0.160
-16.000
-8.190
-0.800
1.600
-1.080
-1.630
-6.000
-2.780
4.460
-2.020
0.930
1.260
-0.310
-9.000
1.100
-0.630
-0.460
s(r) % = range(mean)/1.128 =
Page 38 of 41
2.068
2.321
0.592
1.730
8.449
0.497
0.189
2.820
3.715
2.732
7.249
2.644
0.547
1.159
20.150
3.483
4.713
1.155
3.936
7.198
16.210
5.869
4.048
5.412
3.507
0.596
4.455
1.577
2.079
1.432
4.0843
3.62
= mean range (%)
%
Appendix 6: Raw data for oxygen in Section 4.4
Data plotted in Figure 3. “Range” equals the absolute value of the difference
between Result 1 and Result 2.
R e s . 1
m g /L
8 .9 0
8 .9 9
8 .9 0
9 .1 1
8 .6 8
8 .6 0
8 .8 1
8 .0 2
7 .0 5
6 .9 8
7 .1 3
6 .7 9
6 .5 5
4 .6 8
5 .2 8
7 .4 2
7 .6 2
5 .8 8
6 .0 3
6 .3 3
5 .9 0
6 .2 4
6 .0 2
9 .1 3
9 .1 0
8 .5 0
8 .7 3
8 .0 9
7 .5 6
6 .3 0
6 .4 3
7 .2 5
7 .2 8
8 .0 0
8 .3 8
9 .2 3
9 .0 9
9 .3 7
9 .3 8
9 .3 2
8 .4 7
8 .2 7
8 .3 7
8 .0 9
8 .0 5
7 .3 8
7 .4 9
4 .5 2
4 .4 5
4 .2 9
m e a n
R e s . 2
m g /L
8 .9 1
9 .0 1
8 .9 0
9 .1 2
8 .6 4
8 .5 1
8 .8 1
8 .0 0
7 .0 8
7 .0 1
7 .1 6
6 .7 8
6 .5 3
4 .6 8
5 .3 3
7 .4 0
7 .6 3
5 .8 8
6 .0 6
6 .3 3
5 .9 0
6 .2 7
6 .0 2
9 .1 1
9 .1 4
8 .4 4
8 .7 1
8 .0 9
7 .5 8
6 .3 2
6 .4 4
7 .3 4
7 .3 1
8 .0 3
8 .2 9
9 .2 9
9 .0 8
9 .3 6
9 .3 7
9 .2 5
8 .4 9
8 .2 8
8 .3 1
8 .1 5
8 .0 3
7 .4 0
7 .4 9
4 .4 9
4 .4 4
4 .2 7
m e a n ra n g e :
r a n g e /1 .1 2 8 :
R a n g e
m g /L
0 .0 1
0 .0 2
0 .0 0
0 .0 1
0 .0 4
0 .0 9
0 .0 0
0 .0 2
0 .0 3
0 .0 3
0 .0 3
0 .0 1
0 .0 2
0 .0 0
0 .0 5
0 .0 2
0 .0 1
0 .0 0
0 .0 3
0 .0 0
0 .0 0
0 .0 3
0 .0 0
0 .0 2
0 .0 4
0 .0 6
0 .0 2
0 .0 0
0 .0 2
0 .0 2
0 .0 1
0 .0 9
0 .0 3
0 .0 3
0 .0 9
0 .0 6
0 .0 1
0 .0 1
0 .0 1
0 .0 7
0 .0 2
0 .0 1
0 .0 6
0 .0 6
0 .0 2
0 .0 2
0 .0 0
0 .0 3
0 .0 1
0 .0 2
0 .0 2 6
0 .0 2 4
Page 39 of 41
Appendix 7: Raw data for BOD in example A and B
Results in mg/L O2 consumption. The certified value of the CRM is 206 ± 5 mg/L.
As the average of two results is always reported for ordinary samples, the s is also
calculated from the average of each sample pair in the internal quality control.
Date
12-09-00
01-03-01
13-03-01
02-04-01
14-08-01
05-09-01
19-09-01
16-10-01
07-11-01
28-11-01
11-12-01
13-12-01
15-01-02
22-01-02
30-01-02
11-02-02
06-03-02
18-09-02
02-10-02
Page 40 of 41
Res. 1
218.90
206.46
221.18
215.00
194.96
218.65
223.86
215.58
196.26
210.89
228.40
206.73
207.00
224.49
201.09
218.83
216.69
206.36
215.21
Res. 2
Average
214.77
216.84
220.83
213.65
210.18
215.68
206.50
210.75
218.03
206.50
216.55
217.60
212.19
218.03
213.01
214.30
214.93
205.60
206.89
208.89
222.73
225.57
229.03
217.88
208.47
207.74
213.66
219.08
214.07
207.58
223.13
220.98
218.22
217.46
227.96
217.16
226.18
220.70
Average: 214.84
s:
5.58
s%:
2.60
Appendix 8: Estimation of standard deviation from range
Number of
samples
n=2
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n=10
Factor ,
d2
1.128
1.693
2.059
2.326
2.534
2.704
2.847
2.970
3.078
Estimation of standard deviation
from range (max-min),
/1/ and /13, page 11/.
The standard deviation, s
can be estimated from
s=
range
d2
For comparison
Rectangular
interval
95 % conf. limit.
3.464
3.92
(Example, see Appendix 5 and 6)
Page 41 of 41
TECHNICAL REPORTS FROM EXPERT GROUP QUALITY AND METROLOGY
351
Guðmundsson, H., Calibration and uncertainty of XRF measurements in the SEM. Espoo 1997. Nordtest, NT
Techn Report 351. 42p. NT Project No. 1280-96.
352
Rydler, K.-E. & Nilsson, H., Transparency of national evaluation methods of accreditation in the field of electrical
measurements. Espoo 1997. Nordtest, NT Techn Report 352. 61p. NT Project No. 1255-95.
353
Tambo, M. & Søgaard, T., The determination of gas density - part 1: State of the art in the Nordic countries.
Espoo 1997. Nordtest, NT Techn Report 353. 19p. NT Project No. 1254-95.
354
Tambo, M. & Søgaard, T., The determination of gas density - part 2: Intercomparison - Calibration of gas density
meters with nitrogen. Espoo 1997. Nordtest, NT Techn Report 354. 91 p. NT Project No. 1254-95.
355
Tambo, M. & Søgaard, T., The determination of gas density - part 3: A guideline to determination of gas density.
Espoo 1997. Nordtest, NT Techn Report 353. 29p. NT Project No. 1254-95.
357
Källgren, H., Laboratory weighing - Procedures according to OIML international;
Weights of classes E1, E2, F1, F2, M1, M2, M3 - Part 1: Testing procedures for weights, - Part 2: Pattern evaluation
report. Espoo 1997. Nordtest, NT Techn Report 357. 82p. NT Project No. 1249-95.
379
Ravn, T., Pitkäniemi, M., Wallin, H., Bragadóttir, M., Grønningen, D. & Hessel, H., Checking of UV/VIS
spectrophotometers. Oslo 1998. Nordic committee on food analysis, NMKL Procedure No.7 (1998). 14 p. NT
Project No. 1339-96.
403
Holmgren, M., Observing validation, uncertainty determination and traceability in developing Nordtest test
methods. Espoo 1998. Nordtest, NT Techn Report 403. 12 p. NT Project No. 1277-96.
418
Views about ISO/IEC DIS 17025 - General requirements for the competence of testing and calibration
laboratories. Espoo 1999. Nordtest, NT Techn Report 418. 87 p. NT Project No. 1378-98.
419
Virtanen, V., Principles for measuring customers satisfaction in testing laboratories. Espoo 1999. Nordtest, NT
Techn Report 419. 27 p. NT Project No. 1379-98.
420
Vahlman, T., Tormonen, K., Kinnunen, V., Jormanainen, P. & Tolvanen, M., One-site calibration of the
continuous gas emission measurement methods at the power plant. Espoo 1999. Nordtest, NT Techn Report
420. 18 p. NT Project No. 1380-98.
421
Nilsson, A. & Nilsson, G., Ordering and reporting of measurement and testing assignments. Espoo 1999.
Nordtest, NT Techn Report 421. 7 p. NT Project No. 1449-99.
430
Rasmussen, S.N., Tools for the test laboratory to implement measurement uncertainty budgets. Espoo 1999.
Nordtest, NT Techn Report 430. 73 p. NT Project No. 1411-98.
431
Arnold, M., Roound robin test of olfactometry. Espoo 1999. Nordtest, NT Techn Report 431. 13 p. NT
Project No. 1450-99.
429
Welinder, J., Jensen, R., Mattiasson, K. & Taastrup, P., Immunity testing of integrating instruments. Espoo 1999.
Nordtest, NT Techn Report 429. 29 p. NT Project No. 1372-97.
443
Guttulsrød, G.F, Nordic interlaboratory comparison measurements 1998. Espoo 1999. Nordtest, NT Techn
Report 443. 232 p. (in Dan/Nor/Swed/Engl) NT Project No. 1420-98.
452
Gelvan, S., A model for optimisation including profiency testing in the chemical laboratories. Espoo 2000.
Nordtest, NT Techn Report 452. 15 p. NT Project No. 1421-98.
537
Magnusson, B., Näykki, T., Hovind, H. & Krysell, M., Handbook for calculation of measurement uncertainty in
environmental laboratories. Espoo 2003. Nordtest, NT Techn Report 537. 41 p. NT Project No. 1589-02.
NORDTEST
TECHNICAL REPORT 537
Nordtest, founded in 1973, is an institution under the Nordic Council of
Ministers and acts as a joint Nordic body in the field of conformity
assessment. The emphasis is on the development of Nordic test methods
and on Nordic co-operation concerning conformity assessment. The main
task is to take part in the development of international pre-normative
activity. Nordtest is yearly funding projects in its field of activity.
Nordtest endeavours to
• promote viable industrial development and industrial competitiveness, remove technical barriers to trade and promote the concept
“Approved Once Accepted Everywhere” in the conformity assessment area
• work for health, safety, environment in methods and standards
• promote Nordic interests in an international context and Nordic participation in European co-operation
• finance joint research in conformity assessment and the development and implementation of test methods
• promote the use of the results of its work in the development of
techniques and products, for technology transfer, in setting up standards and rules and in the implementation of these
• co-ordinate and promote Nordic co-operation in conformity assessment
• contribute to the Nordic knowledge market in the field of conformity assessment and to further development of competence among
people working in the field
12
Download
Related flashcards
Create Flashcards